A060280 Number of orbits of length n under the map whose periodic points are counted by A001350.
1, 0, 1, 1, 2, 2, 4, 5, 8, 11, 18, 25, 40, 58, 90, 135, 210, 316, 492, 750, 1164, 1791, 2786, 4305, 6710, 10420, 16264, 25350, 39650, 61967, 97108, 152145, 238818, 374955, 589520, 927200, 1459960, 2299854, 3626200, 5720274, 9030450, 14263078, 22542396
Offset: 1
Examples
a(7)=4 since the 7th term of A001350 is 29 and the 1st is 1, so there are (29-1)/7 = 4 orbits of length 7. x + x^3 + x^4 + 2*x^5 + 2*x^6 + 4*x^7 + 5*x^8 + 8*x^9 + 11*x^10 + 18*x^11 + ...
Links
- Seiichi Manyama, Table of n, a(n) for n = 1..4000
- Michael Baake, Joachim Hermisson, and Peter Pleasants, The torus parametrization of quasiperiodic LI-classes J. Phys. A 30 (1997), no. 9, 3029-3056.
- Michael Baake, John A.G. Roberts, and Alfred Weiss, Periodic orbits of linear endomorphisms on the 2-torus and its lattices, arXiv:0808.3489 [math.DS], Aug 26, 2008. [_Jonathan Vos Post_, Aug 27 2008]
- Larry Ericksen, Primality Testing and Prime Constellations, Šiauliai Mathematical Seminar, Vol. 3 (11), 2008. Mentions this sequence.
- N. Neumarker, Realizability of Integer Sequences as Differences of Fixed Point Count Sequences, JIS 12 (2009) 09.4.5.
- Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
- T. Ward, Exactly realizable sequences
Programs
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Magma
A060280:= func< n | n le 2 select 2-n else (&+[Lucas(d)*MoebiusMu(Floor(n/d)) : d in Divisors(n)])/n >; [A060280(n): n in [1..50]]; // G. C. Greubel, Nov 06 2024
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Maple
A060280 := proc(n) add( numtheory[mobius](d)*A001350(n/d), d=numtheory[divisors](n)) ; %/n; end proc: # R. J. Mathar, Jul 15 2016
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Mathematica
A001350[n_] := LucasL[n] - (-1)^n - 1; a[n_] := (1/n)*DivisorSum[n, MoebiusMu[#]*A001350[n/#]& ]; Array[a, 50] (* Jean-François Alcover, Nov 23 2017 *)
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PARI
{a(n) = if( n<3, n==1, sumdiv( n, d, moebius(n/d) * (fibonacci(d - 1) + fibonacci(d + 1))) / n)} /* Michael Somos, Jan 28 2003 */ -
SageMath
A000032=BinaryRecurrenceSequence(1,1,2,1) def A060280(n): return sum(A000032(k)*moebius(n/k) for k in (1..n) if (k).divides(n))//n - int(n==2) [A060280(n) for n in range(1,41)] # G. C. Greubel, Nov 06 2024
Formula
a(n) = (1/n)* Sum_{d|n} mu(d)*A001350(n/d).
a(n) = A006206(n) except for n=2. - Michael Somos, Jan 28 2003
a(n) = A031367(n)/n. - R. J. Mathar, Jul 15 2016
G.f.: Sum_{k>=1} mu(k)*log(1 + x^k/(1 - x^k - x^(2*k)))/k. - Ilya Gutkovskiy, May 18 2019
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